In this paper, we study TEM-wave propagation inside a hollow coaxial waveguide filled with an inhomogeneous metamaterial composite, with a graded transition between a right-handed material (RHM) and an impedance-matched left-handed material (LHM). The graded transition and the TEM-wave propagation occur in the direction perpendicular to the boundary between the two media, which has been chosen to be the z-direction. The relative permittivity epsilon(omega, z) and permeability mu(omega, z) of the RHM-LHM composite vary according to hyperbolic tangent functions along the z-direction. The exact analytical solutions to Maxwell's equations are derived, and the solutions for the field components and wave behavior confirm the expected properties of impedance-matched RHM-LHM structures. Furthermore, a numerical study of the wave propagation over an impedance-matched graded RHM-LHM interface, using the COMSOL Multiphysics software, is performed. An excellent agreement between the analytical results and numerical simulations is obtained, with a relative error of less than 0.1%. The present method has the ability to model smooth realistic material transitions, and includes the abrupt transition as a limiting case. Finally, the RHM-LHM interface width is included as a parameter in the analytical and numerical solutions, allowing for an additional degree of freedom in the design of practical devices using RHM-LHM composites. Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.